Norris finished second in the Spanish Grand Prix after starting on pole, and has now recorded six successive podium results.
That includes a breakthrough win at the Miami Grand Prix.
Norris moved into second in the drivers’ championship after his Spanish GP result and sits 69 points behind Verstappen.
It’s the first time the Englishman has sat so high in the standings but he is more worried about the deficit to the points leader.
“Even though I moved into second in the championship, that doesn’t matter,” he argued.
“I couldn’t care if I was second or tenth. It’s more about the gap to what Max is and he’s still extending it at the minute.
“That’s something we can’t afford to do or can’t afford to kind of let him run away with it at this point of the season.”
Norris has had opportunities to close the gap between them.
Starting from pole in Barcelona, a sluggish start meant he lost track position to both Verstappen and George Russell – the latter costing him time in the first stint that would ultimately prove critical.
In Canada, a delay in making a decision to pit under a Safety Car also cost him, handing a potential win to Verstappen.
“We should have done better today,” Norris said of his Spanish performance.
“We should have got some points back on Max.
“Potentially, there was a chance to beat him in Canada, so two races that I finished second and he’s won.
“If I just made some better decision in Canada and if I had a better start today, we could have won two races.
“I know there’s a lot of, and there kind of always has been, a lot of ‘shoulda, woulda, couldas’, but we have what it takes.
“It’s just about putting it all together.”
A reversal in the pairs positions in Canada and Spain would see the current points gap at 55-points.
After 10 rounds, Norris is the only driver to have scored in every race this season.
F1 heads to Austria this weekend, the scene of the 24-year-old’s first F1 podium in 2020.
“I’m confident,” he said of his chances at the Red Bull Ring.
“Every weekend we go into now, the car’s performing extremely well, we’re always there or thereabouts within a couple of tenths of pole, and that’s all we can ask for.
“I think we need to bring something a little bit more just to make our life a bit easier.
“It’s a very different layout again; high-speed, I think we have a bit to work on, comparing to Red Bull – Red Bull seem definitely a bit higher, better in high-speed corners than we are.
“Potentially, we’re lacking a touch in that area, but the rest of it is strong.
“It’s been one of my best tracks in terms of my own competitiveness and my most successful tracks so excited to see all the papaya and the grandstands and have a good weekend.”
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